The risks are high: One or both of the boys may die, and even if they survive, some brain damage is inevitable.
CBS News Correspondent Tony Guida says the principal risks in this operation are blood loss and swelling of the brain.
By mid-afternoon, doctors were entering a critical stage as neurosurgeons prepared to separate the intricate web of blood vessels the twins share, said Dr. Jim Thomas, chief of critical care medicine at Children's Medical Center Dallas.
"This is the phase that if successful will allow for the physical separation of the twins," Thomas said.
Thomas relayed a message from the operating room from Dr. Dale Swift, one of five pediatric neurosurgeon involved: "Everything is going fine. There have been no problems."
Doctors have spent more than a year planning the surgery to separate the boys joined at the crown of the head. The operation is expected to take a team of 50 to 60 medical personnel anywhere from 18 to 90 hours to finish.
While each boy has his own brain, they share an extensive attachment of blood vessels. Doctors have said that the risks of the surgery include death and brain damage.
It took doctors about two hours in the morning to position the twins in a specially made bed that lets doctors swivel their bodies for easy access to the front and back of their heads.
Surgeons have removed tissue from the boys' hips that will be used to cover their brains after they are separated. Thomas said surgeons were preparing to remove the skin expanders from the boys' scalp. The expanders were inserted about five months ago to create extra skin. Tissue formed by the expanders will cover the head wounds.
On the right side of each boy's brain, the blood flows in a normal fashion - into the brain and back to the boy. On the left side, though, the blood flows from one boy into the other.
"During the operation, the left hemisphere is going to be at risk," Swift said earlier this week.
He said the hope is that the blood will drain into other deep veins. The worst possible situation, Swift said, is that the blood can't get out and the hemisphere becomes swollen and damaged.
"We think the result will be somewhere in the middle," he said.
Even when the boys are eventually separated, there will still be much work to get done.
"The moment they're separated may seem like a great moment, but it's not," Swift said. "We can't stop doing anything. There's no relaxing right then and there because you have to get this all closed up."
After the surgery, the boys will go to an intensive care unit, where they will remain in a drug-induced coma for three to five days, Swift said.
The boys were born June 2, 2001, by Caesarean section to Sabah Abu el-Wafa and her husband, Ibrahim Mohammed Ibrahim.
Correspondent Guida notes there was a one in 10 million chance they would be born with their deformities.
Dallas-based World Craniofacial Foundation, a nonprofit group that helps children with deformities of the head and face, arranged to bring the boys to Dallas in June 2002 for an evaluation.
A team of specialists determined that the boys could be separated, though the risks include brain damage and death. The boys' father told doctors he felt it was worth it to give them a chance at a normal life.
"If they're left this way, they're not going to be normal," he said.
The father spent much of the past year in Dallas with the boys before returning to Egypt this summer. He returned this week with his wife and the twins' 5-year-old brother, Mahmoud.
The mother and brother had not seen the boys' in 16 months, according to the hospital. The family, along with two Egyptian nurses who have cared for the twins, were in a private area Saturday at the hospital.
"The parents I think are doing fine," Thomas said Saturday afternoon. "I think the parents are helped in may ways by a very strong faith structure they have."
Relatives of the boys in the tiny village of el-Homr, near the southern Egyptian city of Qus, prayed for their safe return.
"We are optimistic but we are very worried about the twins after the failure of the separation of the Iranian twins" in July, said Saad Abu Qaeis, the uncle of the twins' mother. The 29-year-old Iranians died after separation surgery in Singapore.
There have been at least five surgeries around the globe in the past three years to separate twins joined at the head. Three were successful; one resulted in one twin dying and in another both twins died.
Thomas acknowledged the risks but said ethics boards at two hospitals had reviewed the case.
"Nobody goes into this lightly," said Swift, the pediatric neurosurgeon. "At this point, we think it's the right thing to do."